TITLE: The chance of the word to be of all different letters?
QUESTION [6 upvotes]: A four letter word is written down by taking letters from the word KANYAKUMARI.What is the chance of the word to be of all different letters?

Since KANYAKUMARI has K,K,A,A,A,N,Y,U,M,R,I letters in which there are 2 K's and 3 A's.The chance that the four letter word to be of all different letters$=\frac{m}{n}$,where $m=\binom{8}{4}$,because there are 8 different letters K,A,N,Y,U,M,R,I.and $n=\binom{11}{4}$.So probability$=\frac{7}{33}$,but the correct answer is $\frac{840}{1109}$.I dont know where i have made the mistake.Please help me.

REPLY [2 votes]: Following Andre Nicolas's suggestion, we think of the letters as being on 11 "Scrabble" tiles, with all combinations of tiles equally likely, and with the tiles for the A's (and K's) painted with different colors.
Since each selection of 4 tiles can be arranged in $4!$ ways, we can disregard the ordering of the tiles, 
so we get the probability that all 4 letters are distinct is given by
$\hspace{.3 in}\displaystyle\frac{\binom{6}{4}+2\binom{6}{3}+3\binom{6}{3}+6\binom{6}{2}}{\binom{11}{4}}=\frac{205}{330}=\frac{41}{66}$
by considering the cases 
$\;\;\;\;$1) no A's, no K's $\;\;$2) 1K, no A $\;\;$ 3) 1 A, no K $\;\;$ 4) 1A, 1K